Questions: What is the surface area of this triangle in square yards?

Solution Steps

The shape is a triangular pyramid (tetrahedron) with a base that is an equilateral triangle. The given dimensions are:

• Each side of the base triangle: 4 yards
• Height of the pyramid: 3.5 yards
• Slant height of the triangular faces: 3 yards

The base is an equilateral triangle with side length 4 yards. The area \( A \) of an equilateral triangle is given by: \[ A = \frac{\sqrt{3}}{4} \times \text{side}^2 \] \[ A = \frac{\sqrt{3}}{4} \times 4^2 \] \[ A = \frac{\sqrt{3}}{4} \times 16 \] \[ A = 4\sqrt{3} \text{ square yards} \]

Each triangular face of the pyramid is an isosceles triangle with a base of 4 yards and a slant height of 3 yards. The area \( A \) of one triangular face is given by: \[ A = \frac{1}{2} \times \text{base} \times \text{height} \] \[ A = \frac{1}{2} \times 4 \times 3 \] \[ A = 6 \text{ square yards} \]

The total surface area of the pyramid is the sum of the area of the base and the areas of the three triangular faces: \[ \text{Total Surface Area} = \text{Area of Base} + 3 \times \text{Area of One Triangular Face} \] \[ \text{Total Surface Area} = 4\sqrt{3} + 3 \times 6 \] \[ \text{Total Surface Area} = 4\sqrt{3} + 18 \]

Final Answer